Less fighting than expected experiments with wars of attrition and all-pay auctions

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1 Less fighting than expected experiments with wars of attrition and all-pay auctions Hannah Hörisch Oliver Kirchkamp 27th June 2009 Abstract While all-pay auctions are experimentally well researched, we do not have much laboratory evidence on wars of attrition. This paper tries to fill this gap. Technically, there are only few differences between wars of attrition and all-pay auctions. Behaviourally, however, we find striking differences: As many studies, our experiment finds overbidding in all-pay auctions. In contrast, in wars of attrition we observe systematic underbidding. We study bids and expenditure in different experimental frames and matching procedures and tie in with the literature on stepwise linear bidding functions. Keywords: War of attrition, dynamic bidding, all-pay auction, stabilisation, volunteer s dilemma, experiment JEL classification C72, C92, D44, E62, H30 Universität Bonn University of Jena; School of Economics; Carl-Zeiß-Str. 3; Jena; Germany. Fax: , Phone: , An earlier version of this paper circulated under the title Why are stabilisations delayed an experiment with an application to all pay auctions. We thank Sarah Volk for valuable support during the preparation of this paper and the experiment, Radosveta Ivanova-Stenzel, Hartmut Kliemt, and two anonymous referee for helpful comments on earlier versions, and Urs Fischbacher (2007) for ztree. Financial support from the Deutsche Forschungsgemeinschaft through SFB 504 is gratefully acknowledged.

2 1 Introduction Numerous experiments explore models of a static conflict or competition such as first-price all-pay auctions and rent-seeking contests. The laboratory evidence documents that human decision makers tend to overbid and make investments that are even higher than the already high and socially wasteful equilibrium values. This paper investigates which of the properties of the static first-price all-pay auction carry over to the war of attrition, i.e. a second-price all-pay auction. As a benchmark we replicate the standard first-price all-pay auction experiment in a situation with a single prize and two bidders. As expected, we observe some overbidding. Similar to Müller and Schotter (2007) we also find bifurcated bidding functions, i.e. stepwise linear bidding functions. We extend the experimental literature by investigating a war of attrition, which we implement both as a dynamic and as a static bidding process. In contrast to the literature on all-pay auction we find underbidding, in particular in the dynamic implementation of the war of attrition. Furthermore, we do not find bifurcated bidding functions in the war of attrition. A war of attrition can be interpreted as a fight between two players about a prize. For each second during which players are fighting both players incur a cost. As soon as one player gives up, the conflict ends and the other player earns the prize. Both players have to make their investments. Due to its dynamic character the war of attrition has been used as a model of a wide range of problems: Arms races (Zimin and Ivanilov, 1968), fights between animals (Bishop, Canning, and Smith, 1978; Riley, 1980), the voluntary provision of public goods (Bliss and Nalebuff, 1984; Bilodeau and Slivinski, 1996), competition between firms (Ghemawat and Nalebuff, 1985; Fudenberg and Tirole, 1986; Roth, 1996), the settlement of strikes (Kennan and Wilson, 1989; Card and Olson, 1995), fiscal and political stabilisations (Alesina and Drazen, 1991; White, 1995), the timing of exploratory oil drilling (Hendricks and Porter, 1996), and many others are all wars of attrition. Wars of attrition with two players are isomorphic to (static) second-price allpay auctions. In a second-price all-pay auction with two bidders both bidders submit a bid (this bid is equivalent to the amount of time bidders are willing to 1

3 hold out during the war of attrition), the bidder with the highest bid wins the prize, and both bidders pay the amount of the second highest bid. Also in the war of attrition the winner does not have to pay more once the loser gave up, i.e. pays effectively the second highest bid. ½ Wars of attrition are also similar to first-price all-pay auctions. A first-price allpay auction (which we will simply call all-pay auction in the following) is a static procedure where bidders commit to bids or investments previous to the game. The highest bid wins and all bidders pay their own bid. While there are many experiments on all-pay auctions there are surprisingly few on wars of attrition. Our paper tries to fill this gap. We will show that behaviour is quite different under the two institutions. Table 1 on page 3 provides an overview of the experimental literature. Experiments with all-pay auctions have been done by Davis and Reilly (1998), Potters, de Vries, and van Winden (1998), Barut, Kovenock, and Noussair (2002), Gneezy and Smorodinsky (2006), and Müller and Schotter (2007). Four of these studies find a significant amount of overbidding while one (Potters, de Vries, and van Winden, 1998) finds no significant deviation from equilibrium. A famous variant of an all-pay auction is the dollar auction game (Shubik, 1971). Again, overbidding is a standard finding in laboratory experiments. Müller and Schotter (2007) observe what they call bifurcation of bidding functions in experiments with all-pay auctions. Compared with equilibrium bids bids are too small if bidding is expensive, and too large if bidding is cheap. Bidding functions can be approximated with the help of a stepwise linear function similar to the one in figure 1. In section 4.4 we will try to replicate Müller and Schotter s results and check whether these bifurcated bidding functions can also be found in a war of attrition. Another related auction format is a rent-seeking contest. As in all-pay auctions also in rent-seeking contests all players pay their own bid. In contrast to the allpay auction the highest bidder in the rent-seeking contest does not win the auction with certainty. Instead, the probability of winning is only an increasing function of the bid. ¾ Rent-seeking contests have been introduced by Bishop, Canning, and ½ With more than two players the two formats are no longer isomorphic, though, weakly equivalent. In the experiment we present below we will concentrate on the situation with two players. ¾ In Krishna and Morgan (1997) only all-pay auctions in which the highest bidder obtains the prize with certainty are called all-pay auctions. 2

4 Table 1 Comparison with other experiments 3 institution bidding procedure asymmetric information about Millner and Pratt (1989) rent-seeking static overbidding Shogren and Baik (1991) rent-seeking, framed in expected payoffs number of contestants number of prizes repetitions in a group periods static result close to equilibrium Davis and Reilly (1998) rent-seeking first-price static overbidding Potters, de Vries, and van Winden (1998) rent-seeking overbidding in the static first-price rent-seeking case Vogt, Weimann, and Yang (2002) rent-seeking sequential efficient equilibrium selected Barut, Kovenock, and Noussair (2002) first-price static valuation 6 2 or or 50 overbidding Anderson and Stafford (2003) rent-seeking cost, no. of static with entry-fee contestants overbidding Schmidt, Shupp, Swope, and Cardigan (2004) rent-seeking static overbidding Schmidt, Shupp, and Walker (2004) rent-seeking static 4 1,3, 1 1 underbidding Bilodeau, Childs, and Mestelman (2004) volunteer s dilemma dynamic overbidding Öncüler and Croson (2005) risky rent-seeking static 2, overbidding Gneezy and Smorodinsky (2006) first-price static 4,8, overbidding Abbink, Brandts, Herrmann, and Orzen (2007) rent-seeking static overbidding Müller and Schotter (2007) first-price static cost 4 1,2 1,50 50 overbidding, bifurcation underbidding our experiment second-price dynamic in the sequential format, no cost 2 1 1,6 24 first-price static bifurcation

5 Figure 1 A bifurcated bidding function (solid) and an equilibrium bidding function (dashed) β(c) First-price all-pay β A (c i ) c c c i Smith (1978) and Tullock (1980). Possibly the first test of Tullock s model in the laboratory is an experiment by Millner and Pratt (1989). Their, perhaps disappointing, result is that participants make substantially larger bids, i.e. socially wasteful investments, in the rent-seeking contest than they should in equilibrium. Several other experiments with rent-seeking contests followed. Seven of the ten studies of rent-seeking contests find a significant amount of overbidding, two studies (Shogren and Baik, 1991; Vogt, Weimann, and Yang, 2002) find no significant deviation from equilibrium, and only one (Schmidt, Shupp, and Walker, 2004) finds significant underbidding in rent-seeking contests. A generalised war of attrition, or a volunteer s dilemma, is an nth price all-pay auction with n 1 > 1 prizes. A simple version of such a game has been studied experimentally by Diekmann (1993). Participants decide whether to provide a public good. Each player can only choose between two different bids, a low bid (waiting for somebody else to provide the public good) or a high bid (provision of the public good). Bilodeau, Childs, and Mestelman (2004) study experiments of a sequential version of this game where players are fully informed about each other s bidding costs and find overbidding. Following Tullock all-pay auctions can be seen as a special case of rent-seeking contests. Shogren and Baik (1991), Davis and Reilly (1998), Potters, de Vries, and van Winden (1998), Vogt, Weimann, and Yang (2002), Anderson and Stafford (2003), Schmidt, Shupp, Swope, and Cardigan (2004), Schmidt, Shupp, and Walker (2004), Öncüler and Croson (2005), Abbink, Brandts, Herrmann, and Orzen (2007). See Bulow and Klemperer (1999) for a derivation of the equilibrium of such a game. 4

6 In sum, most of the experiments with variants of all-pay auctions find systematic overbidding, i.e. bids that are higher than the risk neutral equilibrium. Actually, even in regular (winner-only-pays) second-price and first-price auctions overbidding is a common phenomenon, although to a lesser degree. If we extrapolate from all-pay auctions and rent-seeking contests, we should expect some overbidding in wars of attrition, too. We will proceed as follows: In section 2 we derive the equilibrium bidding functions. In section 3 we discuss the experimental procedure and present our setup. Section 4 presents the results and section 5 concludes. 2 Hypotheses and Equilibrium predictions 2.1 Equilibrium in the war of attrition In this section we derive the risk neutral Bayesian Nash equilibrium for the war of attrition with two bidders, i and j. As noted above, wars of attrition with two players are isomorphic to second-price all-pay auctions. We will call P the prize from winning the war of attrition. In the auction bidders make bids b by staying for up to b seconds in the auction unless the opponent leaves the auction earlier. If bidder i is staying for b seconds in the auction this bidder incurs a cost b c i, where c i is private information of bidder i and not known to bidder j. The other bidder only knows that c i is drawn from a uniform distribution over [c c] where c is positive and should not be too small compared to c to ensure that the participation constraint is fulfilled. We follow a standard approach and assume that the opponent of bidder i, bidder j with cost per second c j, uses a decreasing bidding function β W e (c) with inverse β W( 1) e ( ). Bidder i does not know c j, but can calculate the expected utility Í i, given that i s bidding cost per second are c i and bidder i bids up to b seconds. If bidders maximise a utility function u(x) See Cox, Smith, and Walker (1988); Kagel, Harstad, and Levin (1987); Kagel and Levin (1993). 5

7 the expected utility is β W e ( 1) (b) c Í i = u( b c i ) f(c j ) dc j + u ( P β W e (c ) j) c i f(cj ) dc j c β }{{} W ( 1) e (b) }{{} bidder i does not win and pays the own bid b bidder i wins P and pays the second highest bid β W e º (1) We will first analyse the risk neutral case. Thus, we can replace u(x) by x, take the first derivative Í i / b and substitute b = β W e (c i) to obtain the first order condition c i ci c f(c j )dc j P f(c i) β W e (c i ) = 0 º (2) If c i is uniformly distributed over [c c] then (2) can be simplified: β W e P (ci ) = c i (c i c) (3) which, with the restriction β W e ( c) = 0, yields the candidate equilibrium bidding function: β W e (c i) = P c ÐÒ ( c c)c i (c i c) c To check whether the individual participation constraint is fulfilled we substitute (4) into (1) and obtain Í i = P c c i + ÐÒ c i c c c (4) º (5) Hence, as long as c is not too small the expected utility from participating is positive for all c i [c c]. This is the case for all combinations of parameters we use in our experiment (see table 2 below). Hence, the expression given by equation (4) describes an equilibrium bidding function. From equation (4) we see that a bidder with a very small c i is ready to make a potentially very high bid. It might even be that β W e (c i) c i > P. Since we are in a war of attrition, a bidder with a very small c i will pay the bid of the opponent with the larger c i which is much smaller than β W e (c i ) most of the time. Therefore, as we see in equation (5), the expected utility, ex ante, is still positive. We should also note that, during the bidding process, our bidder does not necessarily want to stop when b c i = P. During the war of attrition the current bid b c i is already 6

8 Figure 2 Equilibrium bidding functions risk neutral bids constant absolute risk aversion war of attrition all-pay auction β e β e β e war r = of attrition first-price all-pay c i r r = = c c c c c The left diagram shows risk neutral equilibrium bidding functions for the war of attrition (solid line) and all-pay auction (dashed lines). The diagram in the middle shows equilibrium bidding functions for the war of attrition for bidders with constant absolute risk aversion and risk aversion parameters r = 2, r = 6, and r = 20. The diagram on the right shows similar equilibrium bidding functions for the all-pay auction c i r r = r = = c c i a sunk cost at any point in time. Only the marginal increment of the bid and the thereby achieved increase in probability to obtain P are relevant. Hence, while it might seem strange that bidders make bids potentially larger than P, (with our parameters) they will make a positive profit ex ante, and they should not care about their sunk investment ex interim. An example of the equilibrium bidding function is shown as the solid line in the left diagram in figure 2. The expected expenditure is R W e = c c 2.2 Equilibrium in the all-pay auction ( ) β W e (c i) 2 c i c c c 1 + ÐÒ c ( c c) dc c 2 i = P (6) ( c c) 2 Similar to the derivation for the war of attrition in equations (1) to (5) we can find the equilibrium bidding function for the all-pay auction. We assume that bidder j with cost c j uses a decreasing bidding function β A e (c) with inverse β A( 1) e ( ). The Many auction theorists call the sum of all bids revenue and use the letter R to denote revenue. Since in many applications of wars of attrition there is no seller and no revenue we use the term expenditure in this paper. 7

9 expected utility of bidder i with per period cost c i who bids up to b periods is β A e ( 1) (b) c Í i = u( b c i ) f(c j ) dc j + u (P b c i ) f(c j ) dc j c β }{{} A ( 1) e (b) }{{} bidder i does not win and pays the own bid b bidder i wins P and pays the own bid b º (7) Again, we will first consider the risk neutral case u(x) = x. We take the first derivative Í i / b and substitute b = β A e (c i) to obtain the first order condition c i ci c c f(c j )dc j c i f(c j )dc j f(c i) c i β A e (c i ) P = 0 º (8) With c i distributed uniformly over [c c] we can simplify (8) to β A e P (ci ) = c i ( c c) (9) which, with the restriction β A e ( c) = 0, yields the candidate equilibrium bidding function: β A e (c i) = P c c ÐÒ c c i º (10) To check whether the individual participation constraint is fulfilled we substitute (10) into (7) and obtain Í i = P ( c c i)( c c) ÐÒ c c i c c º (11) Again, this expression is positive for all c i [c c] as long as c is not too small. For our parameters this is always the case, hence, the expression given by equation (10) describes an equilibrium bidding function. An example of the equilibrium bidding function is shown as a dashed line in the left part of figure 2. As long as c i (c c) we always have β A e (c i) < β W e (c i). The expected expenditure of the all-pay auction is R A e = c c ( ) β A e (c c c 1 + ÐÒ i) c c c dc c i = P ( c c) 2 º (12) 8

10 Comparing with equation (6) we see that the expected expenditure in the two auction formats is the same, R W e = RA e. Krishna and Morgan (1997) point out that this need not be the case if bidding costs are not distributed independently. They show that generally the expenditure in the war of attrition is larger than or equal to expenditure in the all-pay auction. 2.3 Risk aversion In the previous two sections we have derived equilibrium bidding functions under the assumption of risk neutrality. We know from other auction experiments that bidders behaviour can often be well explained by risk aversion. In this section we derive equilibrium bidding functions for risk averse bidders. While we can not find a closed form solution for general utility functions, it is possible to follow the above steps for specific utility functions. In our context, constant relative risk aversion u(x) = x 1 r is not very meaningful since equilibrium payoffs can be positive and negative. Thus, utility is sometimes real and sometimes complex, which is difficult to interpret. Here we consider only the case of constant absolute risk aversion u(x) = e r x. War of attrition function is With constant absolute risk aversion r the equilibrium bidding β WR e (c i ) = c c ) c c (1 e rp ÐÒ ( c c)c i rc (c i c) c (13) and the expected expenditure is R WR e = 1 e rp ( ( c c r ( c c) c c 1 + ÐÒ c )) º (14) c In the risk neutral limit r = 0 these expressions coincide with their risk neutral counterparts given by (4) and (6). For all positive values of r bids and expenditures are decreasing in r and smaller than the risk neutral values. 9

11 First-price all-pay auction β AR e = The equilibrium bidding function is ) c c ( c c) (1 e rp ) ÐÒ c c r ( c e rp c ) c c (( c c i ) e rp + c i c c (15) ( c c) c i and the expected expenditure is R AR e = (c ÐÒ cc + rp )e rp c c c ÐÒ c c r (e rp c c c c) º (16) In the risk neutral limit r = 0 these expressions coincide with their risk neutral counterparts given by (10) and (12). For positive values of r bids are smaller if c i is sufficiently large. This is consistent with the observation of Fibich, Gavious, and Sela (2006). In their paper risk is modelled in a different way, as weak risk aversion, which is a small perturbation of the risk neutral utility function. They also find that with increasing risk their low type, in our model the bidder with a high c i, bids less the more risk averse bidders become. However, the high type, in our model the bidder with a low c i, bids more the larger the amount of risk aversion. The intuition given by Fibich, Gavious, and Sela also applies in our case: Bidders with a high c i will, most likely, not win the auction and just lose their bid. Thus, making a high bid is risky for them. The more risk averse bidders are, the lower is the equilibrium bid. Bidders with a small c i have good chances to win. Making a slightly too small bid can be very risky for them. Thus, the more risk averse bidders become, the higher are the equilibrium bids for bidders with a low c i. The right diagram in figure 2 shows bidding functions for different values of risk aversion r. We see that for most values of c i risk averse bids are below the risk neutral bid. However, if c i is very small, risk averse bids are higher than the risk neutral bid. The expenditure is decreasing in r and smaller than the risk neutral expenditure. For the parameters we are using R AR e decreases more slowly than R WR e, i.e. with risk averse bidders the expected expenditure is larger with all-pay auctions. 10

12 3 Experimental setups We will use table 1 to discuss our setup. Institution: Most experiments summarised in table 1 have been done with all-pay auctions and rent-seeking contests. There is also one experiment with a volunteer s dilemma. In this paper we concentrate on wars of attrition. We use all-pay auctions as a reference point. With our experiment we want to find out whether the seemingly small differences between the two auction formats affect bidding behaviour and bidding anomalies. Another game that is similar to a war of attrition is the sequential volunteer s dilemma study by Bilodeau, Childs, and Mestelman (2004). In this game players have symmetric information about each other s bidding cost. In the only subgame perfect equilibrium of their game the bidding process ends in the first round with a bid of zero, i.e. in equilibrium one does not observe dynamically increasing bids. Bilodeau, Childs, and Mestelman observe substantially higher bids, i.e. again overbidding. We depart from Bilodeau, Childs, and Mestelman in analysing a situation with asymmetric information. The mutual bidding cost is only revealed at the end of the game, i.e. players do not know ex ante who is the weakest bidder. Bidding procedure: Many other auction experiments use a static bidding procedure. In this paper we will compare two procedures: A static and a dynamic procedure. With the static procedure players simply fill in a number on the screen and the computer immediately determines the winner and gains and losses from the game. With the dynamic procedure participants see their bid and the time like an ascending clock on the screen and can, similar to the procedure in the Dutch auction, decide to stop the clock. Unlike in the Dutch auction, the person who stops the clock is the loser of the auction. With two bidders the two processes are isomorphic, but behaviourally there could be a difference. The dynamic procedure applies naturally in the context of the war of attrition. However, if we find Note that, as in the Dutch auction, bidders in a two-player war of attrition do not learn anything unexpected until the auction ends. If a bidder finds it optimal to hold out until a bid b ex ante, no relevant new information is gained ex interim. 11

13 behavioural differences between all-pay auctions and wars of attrition it will be interesting to know more about the reasons for these differences: Does behaviour differ as the result of a change from a static to a dynamic procedure, or as the result of a change in the strategic situation? Hence, we will study the war of attrition both with a dynamic and a static procedure. Asymmetric information: Most of the early experiments with all-pay auctions and rent-seeking contests use symmetric information of players about all aspects of the game. This symmetry simplifies the experiment considerably. Only some of the more recent experiments (see table 1) introduce asymmetric information. In our introduction we mentioned a couple of examples for war of attritions. In these examples the bidding cost, i.e. the strength of the army or the animal or the stength of the R&D department of a firm, the waiting cost during the settlement of a strike, etc. is neither perfectly known to everybody nor is it the same for all contestants. More technically, the rather abstract case of perfect symmetry rules out equilibria in pure strategies. To avoid mixed equilibria as a benchmark solution and to be closer to the applications we have in mind we will assume that bidding costs are private information and are uniformly distributed over some interval [c c]. Most of the treatments we did were based on intervals with c/ c = 1/2 with c between 3º6 and 5º2 (see appendix A). After normalising the bids we did not find any significant differences between these intervals. Hence, we decided to pool treatments with c/ c = 1/2 but different c. Number of contestants and prizes: Most of the experiments that we summarise in table 1 have two bidders. A setup with two bidders is particularly attractive for a comparison of a dynamic with a static game, since with two bidders the static and the dynamic game are isomorphic. We will, hence, have two bidders and one prize in our experiment. In this context we should mention Anderson, Goeree, and Holt (1998) who develop a model of boundedly rational bidders which relates overbidding to the number of bidders. The more bidders there are the more overbidding we should observe in this model. With two bidders we have the smallest possible number 12

14 of bidders, thus we should not expect too much overbidding. Since we are not interested in the absolute level of overbidding but in a comparison of bidding behaviour in different auction formats, the number of bidders is not essential for us. Repetitions: The Bayesian Nash equilibria, which we present in sections 2.1, 2.2, and 2.3, are the equilibria of a game that is played once or, by backward induction, equilibria of a game that is played finitely many times. We will study both situations: a sequence of one shot games with random matching of players after each round, but also games with random matching of players only after a fixed number of repetitions. While the one shot game constitutes a simple benchmark, we find the treatment with repetitions more interesting. Players who have fought over the allocation of a resource in the past will do this again. Regardless of whether we interpret the war of attrition as an arms race, as the provision of public goods, the competition between firms, the settlement of strikes, or as political stabilisation wars of attrition tend to repeat. Hence, most of our experiments are based on the finitely repeated war of attrition. The repeated situation is not only closer to the conflict we actually want to model, but studying repeated wars of attrition also has a technical advantage in the laboratory since the waiting time for other participants is considerably reduced. Nevertheless, we also have one treatment with random rematching of players after each round in sections 4.2 and 4.3. Implementation in the lab: All treatments of the experiment were implemented with the software z-tree (Fischbacher, 2007) and carried out at the experimental laboratory of the SFB 504 at the University of Mannheim. In our experiment we want to study the impact of the number of repetitions, the role of static versus dynamic bidding, and the difference between a war of attrition and an all-pay auction. Our baseline treatment is a (dynamic) war of attrition with 6 repetitions and cost intervals with 2c = c. The other four treatments are variants of the baseline treatment in which only one factor is changed. Table 2 lists the different treatments. Parameters for each session are given in section A of the appendix. In Since bids can have a large variance some wars of attrition will end quickly while others last for a long time. If players are rematched in each period, most players will have to wait most of the time. Otherwise these different bids average out, which speeds up the experiment considerably. 13

15 Table 2 Treatments c/ c repetitions static firstprice participants sessions β e < b The last column shows the relative fraction of bids b that are larger than the equilibrium bid β e. Figure 3 The bidding interface in dynamic treatment round: 2 of 24 remaining time [sec]: 3596 The value of the prize is 100 The cost of the other bidder is between 2.2 and 4.4 per second Your cost is 3.59 per second You are now bidding the following number of seconds for the prize: 4.00 You have, hence, bid the following amount in this auction: To leave the auction, press the bottom right button I stop bidding the remainder of this section we describe how the treatments were implemented. In the treatments with dynamic bidding groups of typically 10 to 14 participants read instructions (see section B in the appendix), answer computerised control questions to check whether they understand the experiment, and are matched randomly in pairs to bid for a prize. During the bidding process participants see information similar to the one shown in figure 3. ½¼ The number of seconds and the bid are updated every second. As soon as one bidder stops bidding the other is declared winner of the auction. In the static treatment, the screen look looks like the one in figure 4. In ½¼ For technical reasons the top right corner of the screen in the experimental interface also showed a remaining time slowly counting down. We set this time to an exorbitantly high value, so it never became binding in any of our experiments. In the unlikely event of remaining time actually counting down to zero the text would change to please make your decision now but bidders would still be able to increase their bids. This, however, never happened in any of our 14

16 Figure 4 The bidding interface in the static treatment round: 2 of 24 remaining time [sec]: 45 The value of the winner s prize is 100 The cost of the other bidder is between 2.2 and 4.4 per second Your cost in this round is 3.59 per second Please enter the amount of seconds or total cost which you are ready to bid Maximal cost to bid to seconds to cost maximal number of seconds 4.00 Continue Figure 5 Feedback given at the end of a round round: 2 of 24 remaining time [sec]: 17 The other bidder has won the auction auction your other length of your other your new balance cost per bidder s the auction cost bidder s profit of second cost per in (total) cost in this your ac- second seconds (total) auction count contrast to the dynamic procedure (figure 3) players make decisions before the actual bidding begins. They either fix a highest cost or a highest number of seconds up to which they are willing to bid. The interface in the experiment automatically converts cost into seconds and vice versa. Once players have made their choice the actual bidding process completes instantaneously. ½½ In both the dynamic and the static treatment participants get feedback similar to the one shown in figure 5. They are asked to copy this information manually to a table on their desk. Some of the feedback information, such as the other bidder s cost, might not always be available to bidders in a natural context. However, during our pilot experiments we saw that presenting the information in a symmetric experiments. ½½ As long as we look at bidders who make a positive bid participants spend on average about the same amount of time with the dynamic and the static decision screen (33.41 seconds in the dynamic treatment and seconds in the static treatment). 15

17 Figure 6 Cumulative distribution of participants total payoffs in the experiment Fn(x) payoff / [Euro] way helps participants to understand the nature of the game. Depending on the treatment participants are either rematched every round or every six rounds to a new random partner. This procedure is repeated until the 24th round. At the end of the experiment participants complete a questionnaire, and receive their earnings from the experiment in sealed envelopes. Sessions last for about 75 minutes. The cumulative distribution of payoffs at the end of the experiment is shown in figure 6. 4 Results 4.1 Bidding Equations (4) and (10) describe equilibrium bids β W e and βa e in the war of attrition and all-pay auction, respectively. The higher the individual cost c i the earlier a participant will give up. The last column of table 2 shows the relative frequency of bids b that are larger than equilibrium bids β e for the different treatments. We can already see that overbidding is more frequent in the all-pay auction and in the static version of the war of attrition while in all dynamic versions of the war of attrition the majority of bids are smaller than equilibrium bids. Figure 7 shows scatterplots of bids in the war of attrition and in the all-pay auction. Let us first have a look at the left graph in figure 7. Each dot corresponds to one bid in the dynamic war of attrition. We also show contour lines of a kernel density estimate. The solid line indicates the equilibrium bid (equation 16

18 Figure 7 Scatterplots of bids war of attrition (dynamic) war of attrition (static) all pay auction b c equilibrium bid P/c 20 b c equilibrium bid P/c b c equilibrium bid P/c c c c c c c The figure shows bids and contour lines of a kernel density estimate from all treatments where c = 2c. Out of the 1788 individual bids in the left graph 64 are out of the range of the graph. Out of the 600 individual bids in the center graph 9 are out of the range of the graph. The graph for the all-pay auction includes also the bids of the winners. We use R s kde procedure with default parameters for the kernel density estimate. (4)). Generally, bidders with a large cost make smaller bids which is in line with the equilibrium prediction. We also see that there is a substantial variance. Furthermore, we find that many bids for the smaller values of c in the war of attrition concentrate around P/c i (dashed line). These bidders leave the auction when they have exactly spent the value of the prize. It also seems that bids follow a different pattern for small and for large values of c. We will come back to this point in section 4.4 when we talk about bifurcated bidding functions. In section 4.4 we will also discuss why the graphs in figure 7 look so differently. Meanwhile, we will provide a more formal comparison of the treatments in the next sections. 4.2 Comparison with equilibrium bids We use an interval regression to estimate the following equation: b i t = β e i (1 + β Û Ö d Û Ö + β Ø Ø d Ø Ø + β ÖÒ d ÖÒ + β ÐÐÔ d ÐÐÔ ) + ν i + u i t (17) where β e i is the equilibrium bid from equation (4) or (10) and b i t is the actual bid. Since we do not observe the winner s bid in the dynamic war of attrition we use an interval regression (see Tobin, 1958; Amemiya, 1973, 1984; Quing and Pierce, 1994). d Û Ö is a dummy that is one in the dynamic war of attrition with 17

19 Table 3 Interval regression random effects estimation of equation (17) β σ z P >t ±ÓÒ º ÒØ ÖÚ Ð β Û Ö º4245 º º415 0º000 º46526 º38375 β Ø Ø º25721 º º609 0º000 º33349 º18093 β ÖÒ º19658 º º209 0º000 º28813 º10504 β ÐÐÔ º68562 º º508 0º000 º47913 º89211 The estimation includes only experiments with c/c = 2. repeated interaction, d Ø Ø is one in the static war of attrition, d ÖÒ is one in the war of attrition in which players are randomly rematched after each period, and d ÐÐÔ is one in the all-pay auction. ν i is a random effect for each subject, u i t is the noise term for each individual choice. If bidders follow the equilibrium bidding function then we should estimate β Û Ö = β Ø Ø = β ÖÒ = β ÐÐÔ = 0. ½¾ Table 3 presents an interval regression random effects model with a random effect for each participant. ½ In the estimation of equation (17) we observe the following: In the all-pay auction there is a significant amount of overbidding (about 69%). This is what we should expect from previous experiments. In all other situations, the dynamic war of attrition, the static war of attrition, and the war of attrition with random rematching we find underbidding (between 20% and 42% depending on the treatment). In particular, underbidding in the war of attrition is always significant. We also ran a regression with an additional dummy which was one if the participant won the auction in the previous period and zero otherwise. The coefficient of such a dummy is small (-0.045) and not significant (p-value of 0.113). Estimates of the ½¾ The ratio between risk averse equilibrium bids and risk neutral bids is (1 e 200 r )/200r for the war of attrition and (200r + ÐÒ2 e 200 r ÐÒ2)/(200 (2 e 200 r ) r (1 ÐÒ2)) for the all-pay auction. Hence, when we compare the all-pay auction with the war of attrition then we can t simply compare levels of these ratios (or levels of estimated coefficients). We can still compare bids in the experiment with risk neutral bids. If we observe a significant amount of underbidding (compared with risk neutral bids) in the war of attrition and a significant amount of overbidding in the all-pay auction then no degree of risk aversion can be consistent with both observations. ½ A model with random effects for each session of the experiment yields very similar coefficients and p-values. A model with fixed effects for participants can not be used here since participants are fully nested in treatments. 18

20 Figure 8 Estimation of equation (17) for different quarters of the experiment β war static rand. allp first second third last Periods [ Quarters of the Experiment ] other coefficients (and their p-values) do not change substantially, we therefore do not provide an extra table for this estimation. To see whether players systematically change their behaviour during the experiment and perhaps converge to the Bayesian Nash equilibrium we divide the periods in the experiment into four parts of equal size, the first, the second, the third, and the last quarter of the experiment. We estimate equation (17) separately for each quarter. Results are shown in figure 8. We see that for all auction formats bids decrease over time in a similar way. The distance between the war of attrition and the all-pay auction remains similar during the course of the experiment. To summarise this subsection: While the experimental literature on all-pay auctions and rent-seeking contests typically finds overbidding our results document underbidding in the war-of-attrition. ½ 4.3 Comparison of expenditures Based on the equilibrium expenditures given by equations (6) and (12) and for our distribution of bidding cost we should expect that the level of expenditure is the ½ Among experimental papers on all-pay auctions and rent-seeking contests only Schmidt, Shupp, and Walker (2004) find significant underbidding in three different types of rent-seeking contests (a random single prize, a random multiple prize and a deterministic proportionate prize contest). The results of Schmidt, Shupp, and Walker (2004) are hard to compare to ours since their setup differs from ours along many dimensions, e.g. the type of game, its static character, the number of contestants. Additionally, in Schmidt, Shupp, and Walker (2004) each participant played the three different contests but each contest was played one-shot, each participant has the same bidding costs and there is no asymmetric information concerning bidding costs. 19

21 Table 4 Expenditure: Results of estimating equation (18) β σ t p value 95% conf interval β Û Ö β Ø Ø β ÖÒ β ÐÐÔ Random effects regression with a random effect for subject and one for matching group. The estimation includes only experiments with c/c = 2. same under the war of attrition and the all-pay auction. For other distributions of the bidding cost this need not be the case. One can show that in equilibrium expected expenditures in the war of attrition are never smaller than in the all-pay auction, they can only be larger (see Krishna and Morgan, 1997). In the experiment we have seen that bids are smaller than equilibrium bids in the war of attrition and larger in the all-pay auction. What can we now say about expenditure? Average overbidding in the all-pay auction does not necessarily imply higher expenditures, at least not if the amount of overbidding depends on the bidding cost. Similarly, average underbidding in the war of attrition does not need to imply lower than equilibrium expenditures. To compare expenditures in the experiment we estimate the following random effects model R i t = R e i t (1 + β Û Öd Û Ö + β Ø Ø d Ø Ø + β ÖÒ d ÖÒ + β ÐÐÔ d ÐÐÔ ) + ν i + ν s + u i t (18) where R i t is the actual expenditure obtained in the auction and R e i t the ex-post expected expenditure in equilibrium. d Û Ö, d Ø Ø, d ÖÒ, d ÐÐÔ, ν i, and u i t have the same interpretation as in equation (17). ν s is a random effect for each session. If all bidders follow the Bayesian Nash equilibrium bidding function we should have β Û Ö = β ÐÐÔ = β ÖÒ = β Ø Ø = 0. If the coefficients are negative the expenditure is smaller than in equilibrium. If the coefficients are positive then expenditure is larger than in equilibrium. Results of the estimation are shown in table 4. ½ We observe the following: For all variants of the war of attrition that we studied (dynamic or static ½ Outliers have been eliminated using Hadi s method (Hadi, 1992, 1994). 20

22 bidding, random or repeated matching) the estimated coefficients are significantly negative, i.e. the expenditure in the war of attrition is smaller than the expenditure in equilibrium. Expenditures in the all-pay auction are larger than in the war of attrition ½ but significantly smaller than equilibrium expenditures. Thus, while expenditures in the war of attrition and the all-pay auction are still socially wasteful in our experiment their level is lower than predicted by equilibrium. This might look surprising since we found bids in the all-pay auction to be larger than equilibrium. The next section might offer an answer to this puzzle. 4.4 Bifurcations in all-pay auctions and in wars of attrition Let us have another look at figure 7. The three scatterplots show bids for the dynamic war of attrition, the static war of attrition, and the all-pay auction. The figure suggests that, in particular for the all-pay auction, bids can be divided into two distinct groups: bids for large values of c and bids for small values of c. The latter are high (and do not depend much on the specific value of c), the former are low (and, again, do not depend much on c). In an experiment with all-pay auctions Müller and Schotter (2007) observe a similar phenomenon and call it bifurcation of bidding functions. If bidding is expensive, bidders underbid. If bidding is cheap, bidders overbid (see figure 1 on page 4). We want to find out whether this is a stable pattern that repeats in our experiments with wars of attrition, i.e. whether individuals with low costs fight longer than predicted by equilibrium and those with high cost stop fighting earlier. The left diagram in figure 7 already suggests that this phenomenon is less pronounced in the war of attrition. However, figure 7 only shows the aggregate distribution. In this section we will look at individual bidding functions. We follow the procedure suggested by Müller and Schotter. If bids can also be divided in the war of attrition, this procedure might be a sensible way to structure ½ A two-sided (parametric bootstrap) test of β Û Ö = β ÐÐÔ yields a p = 0º

23 Figure 9 Stepwise approximation of bidding functions for 18 participants in the war of attrition estimated stepwise bidding function, loss, win The nine diagrams on the left are examples for bidders who can best be approximated with a descending step in the bidding function, the nine on the right are examples for bidders who can best be approximated with an ascending step in the bidding function. the data. Müller and Schotter estimate the following switching regression: b i (c) = { βe i c + β 0 i + u if c ^c i γ e i c + γ 0 i + u if c > ^c i º (19) We can use an OLS regression for the static war of attrition and the all-pay auction. For the dynamic war of attrition we have to use an interval regression (Tobin, 1958; Amemiya, 1973, 1984). For each individual i we estimate coefficients ^β e i ^β 0 i ^γ e i ^γ 0 i. Also for each individual we choose the position of the step ^c i such that the likelihood of the interval regression is maximised. ½ Figure 9 shows several examples of individual bids and estimated individual bidding functions. The nine diagrams on the left show examples for bidders who can best be approximated with a descending step in the bidding function, the nine on the right are examples for bidders who can best be approximated with an ascending step in the bidding function. Müller and Schotter test their approach by comparing the residual sum of ½ For the interval regression we do not obtain convergence of the estimator for 2 of our 282 participants. To check overall convergence we did all our estimates twice, once with at most five iterations, and once with at most 25 iterations. The results are practically the same (we have looked at distributions of estimated coefficients like those presented in figure 10 and found that they are visually indistinguishable for 5 and for 25 iterations) so that we have no reason to believe that a larger number of iterations might change the results. 22

24 Figure 10 Bifurcation, the distribution of i Fn(x) dynamic war static war all pay i The graph shows the cumulative distribution of the step size i. For the static treatments i is given for an OLS approach, for the dynamic treatment i is shown for the interval regression. squares of the switching regression model (19) with the residual sum of squares of an alternative model (which contains the equilibrium bidding function as a special case). We did the same with our data and come to the same conclusion: the switching regression model has a better fit. We should keep in mind that the switching regression model permits all kinds of stepwise functions. Figure 9 suggests that in the war of attrition not all participants use bidding functions that are flat for small bidding costs, then decrease sharply, and then remain flat for high bidding costs. We have to ask ourselves which examples are more representative of bidding in the war of attrition: the ones in the left part of figure 9 or the ones in the right part? To answer this question we will look at the step size, i.e., the difference i = ^β e i^c i + ^β 0 i (^γ e i^c i + ^γ 0 i ) º (20) This difference is positive if participants have a decreasing step in their bidding function like in the left part of figure 9. Figure 10 shows the distribution of the step size. Let us first look at the dotted line which shows the distribution of the step size i for the all-pay auction. As we should expect from Müller and Schotter, more than 50% (actually 59.1%) have an estimated positive stepsize. For the different treatments we test against i = 0 and report results in table 5. For the all-pay auction and the static war of attrition we do not find i to 23

25 Table 5 Two-sided tests against i = 0 i t P > t P Ò dynamic war of attrition (interval regression) static war of attrition (OLS) static all-pay auction (OLS) t-statistics are derived using the Huber-White method to correct for correlated responses from cluster samples. P Ò are p values of a binomial test. Figure 11 Approximating equilibrium bidding functions with stepwise linear functions b war of attrition all-pay auction c c be significantly different from zero. i is actually significantly negative in the dynamic war of attrition where only 36.6% of participants seem to use a positive stepsize. How can it be that in our experiment most participants seem to use a negative stepsize in the dynamic war of attrition treatment? To understand this better let us compare the equilibrium bidding functions in the war of attrition and the all-pay auction. Figure 11 shows examples for the equilibrium bidding functions in the war of attrition (equation (4)) and the all-pay auction (equation (10)) as solid lines. Since Ð Ñ c c β W e (c) = the function is not easy to approximate with a stepwise linear function. The dotted line shows the result of an OLS regression of a random sample of cost values. We see that, since the left part of the function is very steep, we obtain a negative stepsize. The equilibrium bidding function for 24

26 the all-pay auction is much easier to fit with a stepwise linear function (the dotted line is very close to the equilibrium bidding function). Any step of a stepwise linear approximation will be small. 5 Concluding remarks In this series of experiments we address a couple of hypotheses, some related to theoretical studies of the war of attrition, some to other experiments. In line with previous experiments we have found overbidding in all-pay auctions. However, we have seen that overbidding does not carry over to wars of attrition. While equilibrium expenditures are the same in all-pay auctions and in wars of attrition we have found that in the experiment expenditures are lower in wars of attrition than in all-pay auctions. We could see that two effects work handin-hand: First, moving from a first-price format (as in the all-pay auction) to a second-price format (as in the static version of the war of attrition) reduces expenditures. Second, moving from a static format (as in the static version of the war of attrition) to a dynamic format (as in the proper war of attrition) lowers expenditures even more. These results might be interesting for the designer of a contest. If the designer of a contest wants to maximise expenditure (as, e.g., in a competition of students) it might be attractive to choose a first-price format. Contestants should be able to commit to a bid and no information about the number of remaining contestants should be made public. If expenditures are socially wasteful (as in a patent race) we should do the opposite. In line with previous findings of Müller and Schotter (2007) we have found support for discontinuous individual bidding functions in the all-pay auction. For bids in all-pay auctions it matters mainly whether costs are high or low. Participants do not distinguish between intermediate values. With low costs bid are high (and are not much affected by the exact level of the costs), with high costs bids are low (and are, again, not much affected by the exact level of the costs). Also in the war of attrition bidders have a tendency to follow two different regimes, one for low and one for high costs. However, the very simple approximation, which worked 25

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